Question

Prove that ΔABC and ΔEDC are similar.



triangles ABC and DEC where angles A and E are right angles, AC equals 4, AB equals 3, BC equals 5, DC equals 15, DE equals 9, and CE equals 12

15 over 4 equals 12 over 5 equals 9 over 3 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SAS Similarity Postulate.

∠DCE is congruent to ∠BCA by the Vertical Angles Theorem and 15 over 5 equals 12 over 4 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SAS Similarity Postulate.

∠E and ∠A are right angles; therefore, these angles are congruent since all right angles are congruent. 12 over 4 equals 9 over 3 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SSS Similarity Postulate.

∠DCE is congruent to ∠CBA by the Vertical Angles Theorem and 15 over 5 equals 12 over 4 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SSS Similarity Postulate.

Answer 1

Since $\frac{15}{5} = \frac{12}{4}$ shows two corresponding sides are proportional and $\angle DCE \cong \angle BCA$ based on the vertical angles theorem, therefore, we can prove that $\triangle ABC \sim \triangle EDC$ by the SAS Similarity Theorem. (Option B).

Recall:

• Corresponding sides of two triangles that are similar to each other are always proportional.

In the diagram given,

We know that, based on the vertical angle theorem, <DCE and <BCA are congruent.

If we the two sides that the congruent angles lie in between are proportional, then we can prove that both triangles are congruent by the SAS similar theorem.

Thus:

$\frac{DC}{BC} = \frac{EC}{AC}$

• Substitute

$\frac{15}{5} = \frac{12}{4} = 3$ (this shows proportionality)

Therefore, since $\frac{15}{5} = \frac{12}{4}$ shows two corresponding sides are proportional and $\angle DCE \cong \angle BCA$ based on the vertical angles theorem, therefore, we can prove that $\triangle ABC \sim \triangle EDC$ by the SAS Similarity Theorem. (Option B).

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Answer 2

Answer:

B

Step-by-step explanation:

∠DCE is congruent to ∠BCA by the Vertical Angles Theorem and 15 over 5 equals 12 over 4 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SAS Similarity Postulate.